Abstract
Let R be a prime ring of characteristic different from 2, Q r its right Martindale quotient ring and C its extended centroid. Suppose that F, G are generalized skew derivations of R with the same associated automorphism α, and p(x 1, …, x n ) is a non-central polynomial over C such that
for all x, y ∈ {p(r 1, …, r n ): r 1, …, r n ∈ R}. Then there exists λ ∈ C such that F(x) = G(x) = λα(x) for all x ∈ R.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Argaç, L. Carini, V. De Filippis: An Engel condition with generalized derivations on Lie ideals. Taiwanese J. Math. 12 (2008), 419–433.
M. Brešar, C. R. Miers: Strong c…ty preserving maps of semiprime rings. Can. Math. Bull. 37 (1994), 457–460.
J. -C. Chang: On the identity h(x) = af(x) + g(x)b. Taiwanese J. Math. 7 (2003), 103–113.
C. -L. Chuang: Identities with skew derivations. J. Algebra 224 (2000), 292–335.
C. -L. Chuang: Differential identities with automorphisms and antiautomorphisms. II. J. Algebra 160 (1993), 130–171.
C. -L. Chuang: Differential identities with automorphisms and antiautomorphisms. I. J. Algebra 149 (1992), 371–404.
C. -L. Chuang: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103 (1988), 723–728.
C. -L. Chuang: The additive subgroup generated by a polynomial. Isr. J. Math. 59 (1987), 98–106.
C. -L. Chuang, T. -K. Lee: Identities with a single skew derivation. J. Algebra 288 (2005), 59–77.
C. -L. Chuang, T. -K. Lee: Rings with annihilator conditions on multilinear polynomials. Chin. J. Math. 24 (1996), 177–185.
V. De Filippis: A product of two generalized derivations on polynomials in prime rings. Collect. Math. 61 (2010), 303–322.
O. M. DiVincenzo: On the nth centralizer of a Lie ideal. Boll. Unione Mat. Ital., VII. Ser. 3-A (1989), 77–85.
C. Faith, Y. Utumi: On a new proof of Litoff’s theorem. Acta Math. Acad. Sci. Hung. 14 (1963), 369–371.
I. N. Herstein: Topics in Ring Theory. Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, 1969.
N. Jacobson: PI-Algebras: An Introduction. Lecture Notes in Mathematics 441, Sprin- ger, Berlin, 1975.
N. Jacobson: Structure of Rings. American Mathematical Society Colloquium Publica- tions 37, AMS, Providence, 1956.
C. Lanski, S. Montgomery: Lie structure of prime rings of characteristic 2. Pac. J. Math. 42 (1972), 117–136.
J. -S. Lin, C. -K. Liu: Strong c…ty preserving maps on Lie ideals. Linear Alge- bra Appl. 428 (2008), 1601–1609.
C. -K. Liu: Strong c…ty preserving generalized derivations on right ideals. Monatsh. Math. 166 (2012), 453–465.
C. -K. Liu, P. -K. Liau: Strong c…ty preserving generalized derivations on Lie ideals. Linear Multilinear Algebra 59 (2011), 905–915.
J. Ma, X. W. Xu, F. W. Niu: Strong c…ty-preserving generalized derivations on semiprime rings. Acta Math. Sin., Engl. Ser. 24 (2008), 1835–1842.
W. S. Martindale III: Prime rings satisfying a generalized polynomial identity. J. Algebra 12 (1969), 576–584.
E. C. Posner: Derivations in prime rings. Proc. Am. Math. Soc. 8 (1958), 1093–1100.
T. -L. Wong: Derivations with power-central values on multilinear polynomials. Algebra Colloq. 3 (1996), 369–378.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
De Filippis, V. Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings. Czech Math J 66, 271–292 (2016). https://doi.org/10.1007/s10587-016-0255-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-016-0255-0